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Properties

Label 2-1148-7.4-c1-0-16
Degree $2$
Conductor $1148$
Sign $0.897 + 0.441i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.666 + 1.15i)3-s + (−0.883 − 1.53i)5-s + (2.62 − 0.341i)7-s + (0.611 + 1.05i)9-s + (1.24 − 2.15i)11-s − 2.30·13-s + 2.35·15-s + (0.702 − 1.21i)17-s + (−1.97 − 3.42i)19-s + (−1.35 + 3.25i)21-s + (0.0202 + 0.0351i)23-s + (0.939 − 1.62i)25-s − 5.62·27-s + 6.20·29-s + (1.49 − 2.58i)31-s + ⋯
L(s)  = 1  + (−0.384 + 0.666i)3-s + (−0.395 − 0.684i)5-s + (0.991 − 0.128i)7-s + (0.203 + 0.352i)9-s + (0.374 − 0.649i)11-s − 0.638·13-s + 0.608·15-s + (0.170 − 0.294i)17-s + (−0.453 − 0.785i)19-s + (−0.295 + 0.710i)21-s + (0.00422 + 0.00732i)23-s + (0.187 − 0.325i)25-s − 1.08·27-s + 1.15·29-s + (0.268 − 0.464i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435101784\)
\(L(\frac12)\) \(\approx\) \(1.435101784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.62 + 0.341i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.666 - 1.15i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.883 + 1.53i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.24 + 2.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 + (-0.702 + 1.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0202 - 0.0351i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.20T + 29T^{2} \)
31 \( 1 + (-1.49 + 2.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0975 + 0.168i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 1.19T + 43T^{2} \)
47 \( 1 + (-3.71 - 6.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.89 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.75 + 4.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.95 + 3.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.89 - 5.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-6.14 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.679 - 1.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.02T + 83T^{2} \)
89 \( 1 + (-0.113 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.43T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.809674840749998135034801961496, −8.859984283301513040134415712145, −8.197293334178832651463098084337, −7.42111015784704359832892118626, −6.28747244417632457392842294810, −5.02097433670757221738491155966, −4.78974884351206131602528488008, −3.85749552052576051441804068410, −2.32587596706746415087434226839, −0.77576893261718426131715805956, 1.23704729477004771779959746863, 2.35563375242052918892944586614, 3.74624580554643868615382322970, 4.67104480545589936697614705681, 5.74947043194336405696864620829, 6.71954267685543363671978083060, 7.27750848537597204975195730150, 8.016587642981798099810588753050, 8.949536066478995038009504854573, 10.04063718290700775791390329022

Graph of the $Z$-function along the critical line